The Bielefeld Arithmetic Geometry Seminar 2014 (winter term)

Welcome to the website of the 4th edition of the Bielefeld arithmetic geometry seminar! Below you will find a list of the upcoming talks, hover with your mouse over abstract to see the abstract. The seminar is related to projects B5 and C7.
The seminar is supported by the SFB CRC 701: Spectral Structures and Topological Methods in Mathematics.

Talks

Wednesday 22 October, 2014
16:15	D5-153	Veronika Ertl (Universität Regensburg) Overconvergent de Rham-Witt connections [Abstract] For a smooth scheme over a perfect field of characteristic $p>0$, we generalise a definition of Bloch and introduce overconvergent de Rham-Witt connections. This provides a tool to extend the comparison morphisms of Davis, Langer and Zink between overconvergent de Rham-Witt cohomology and Monsky-Washnitzer respectively rigid cohomology to coefficients. In this talk I will describe the main constructions and explain how the comparison theorems can be adapted.
Wednesday 12 November, 2014
16:15	D5-153	Moritz Kerz - Cancelled! Specialization of zero cycles [Abstract] We study the Chow group of zero cycles on algebraic varieties over a local field. In case the local field is a finite extension of $\mathbb Q_p$ one expects certain finiteness properties for this Chow group. We will explain an approach to these conjectures in terms of a specialization map.
Wednesday 26 November, 2014
15:00	D5-153	Daniel Macias Castillo (Instituto de Ciencias Matemáticas, Madrid) Congruences between derivatives of twisted Hasse-Weil $L$-functions [Abstract] The results discussed in this talk include joint work with David Burns, Christian Wuthrich, Werner Bley and Stéphane Viguié. Let $A$ be an abelian variety defined over a number field $k$. Then for any finite Galois extension $F$ of $k$ with group $G$ the equivariant Tamagawa number conjecture (`ETNC') for the pair $(h^1(A_F)(1),\mathbb Z[G])$ was formulated by Burns and Flach as a natural refinement of the seminal conjecture of Bloch and Kato. Under certain not-too-stringent conditions we give a reformulation involving the finite support cohomology of Bloch and Kato of the $p$-part of this conjecture (for a given prime number $p$). We next describe several (conjectural) consequences of this reinterpretation. These concern elements which interpolate the values at $s=1$ of higher derivatives of the Hasse-Weil $L$-functions of twists of $A$ by irreducible complex characters of $G$, suitably normalised by a product of explicit equivariant regulators and periods. They include integrality properties, statements concerning the Galois structure of Tate-Shafarevich groups and integral congruences involving the $G$-valued height pairings of Mazur and Tate. In another direction, we discuss how our approach leads to the first verifications of the $p$-part of the ETNC for elliptic curves $A$ defined over $\mathbb Q$ in the technically most demanding case in which $A$ has strictly positive rank over $F$ and the Galois group $G$ is both non-abelian and of order divisible by $p$.
Wednesday 21 January, 2015
16:15	D5-153	Otmar Venjakob (Universität Heidelberg) Explicit Reciprocity Laws and $(\varphi,\Gamma)$-Modules [Abstract] Various explicit reciprocity laws have been given throughout the history of local class field theory. E.g. for the Hilbert norm residue symbol for example Artin and Hasse found an explicit description which was later generalized by Iwasawa in the context of cyclotomic fields. Later Wiles proved a higher explicit reciprocity law in the context of Lubin-Tate formal groups, which has been generalized by Kato. In the cyclotomic theory one approach is given by the theory of $(\varphi,\Gamma)$-modules as has been developped by Fontaine, Herr, Colmez, Benois, Berger and others. In particular it allows to calculate local Galois cohomology and Iwaswa cohomology rather explicitly. In the talk I am going to report on some developments of the latter in the context of Lubin-Tate formal groups.
Wednesday 28 January, 2015
16:15	D5-153	Massimo Bertolini (Universität Essen) Anticyclotomic main conjectures [Abstract] I will report on recent results on the main conjectures of Iwasawa theory for elliptic curves over the anticyclotomic tower of an imaginary quadratic field. These results are obtained by combining the theory of Euler systems arising from Heegner points with the work of Skinner-Urban.
Wednesday 4 February, 2015
15:00	D5-153	Jiro Nomura (Keio University) The Galois Brumer-Stark conjecture [Abstract] For finite abelian extensions of number fields, the Brumer-Stark conjecture has been studied for many years. This conjecture predicts a deep relation between the special values of $L$-functions attached to the extensions and the Galois module structure of ideal class groups. For the last 5 years, some non-abelian generalizations of the conjecture have been formulated. In this talk, I introduce one of the generalizations, the Galois Brumer-Stark conjecture, which was formulated by Gaelle Dejou and Xavier-François Roblot. Moreover, I prove the conjecture for Galois extensions with specified non-abelian Galois groups.
Tuesday 17 March, 2015
15:00	V3-201	Ma Li (Paris VI) $p$-adic Gross-Zagier formula for Hilbert modular forms [Abstract] In this talk I will present my thesis work on the generalization of Perrin-Riou's $p$-adic Gross-Zagier formula. Let $f$ be a $p$-ordinary Hilbert modular form over a totally real field $F$, and let $E$ be a totally imaginary quadratic extension of $F$. We construct a $p$-adic $L$ function, $L_p$, which interpolates special values of complex $L$ functions associated to $f$ and $E$, and show that the derivative of $L_p$ is, up to a non-zero constant, equal to the $p$-adic height of a Heegner divisor on a Shimura curve over $F$.

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Seminars

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The Bielefeld Arithmetic Geometry Seminar 2014 (winter term)

Talks

Wednesday

22 October, 2014

Wednesday

12 November, 2014

Wednesday

26 November, 2014

Wednesday

21 January, 2015

Wednesday

28 January, 2015

Wednesday

4 February, 2015

Tuesday

17 March, 2015